An overview of the history of projective representations (spin representations) of groups
aa r X i v : . [ m a t h . HO ] D ec An overview of the history ofprojective representations(spin representations) of groups ∗ Takeshi HIRAI*
Contents
A.1 Prehistory 1
A.1.1 Benjamin Olinde Rodrigues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A.1.2 William Rowan Hamilton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A.2 History 8
A.2.1 Creation of theory of representations of groups . . . . . . . . . . . . . . . . . . . . . 8A.2.2 Beginning of projective (or spin) representations of groups . . . . . . . . . . . . . . . 10A.2.3 Cases of Lie groups and Lie algegras . . . . . . . . . . . . . . . . . . . . . . . . . . . 13A.2.4 Spin theory in quantum mechanics and mathematical foundation of quantum mechanics 14A.2.5 Rediscovery of a work of A.H. Clifford . . . . . . . . . . . . . . . . . . . . . . . . . . 18A.2.6 Development in mathematics, and quantum mechanics . . . . . . . . . . . . . . . . . 20A.2.7 Renaissance of the theory of spin representations . . . . . . . . . . . . . . . . . . . . 22A.2.8 Weil representations of symplectic groups . . . . . . . . . . . . . . . . . . . . . . . . 23
To overview the history of spin representations (projective representations) of groups, herewe choose the style to insert necessary explanations in a chronology in such a way that firstwe put age and main items in bold face, and cite main histotical papers, then, explain mainlyits contents and important meaning. (More detailed overview of the history, containing theviews from theoretical physics, was given in [HHoH].) Hoping to give the original flaver as faras possible, we quote original sentances of papers frequently even in German and in French.You can translate them by using website dictionaries if necessary. ∗ This is a self English translation, with small corrections, of Appendix A of my book
Introduction to thetheory of projective representations of groups in Japanese [Hir2]. .1 Prehistory A.1.1 Benjamin Olinde Rodrigues A.1.1.1 Rodrigues’s paper1840 (Virtual discovery of quaternion, and Rodrigues expression of spatialmotions) [Rod] Olinde Rodrigues,
Des lois géométriques qui régissent les déplacements d’un système solidedans l’espace, et la variation des coordonnées provenant de ses déplacements considérés indépendam-ment des causes qui peuvent les produire,
Journal de Mathématiques Pures et Appliquées, (1840),380–440. [Translation of the title] Geometric rules which govern displacements of solid bodies inthe space, and the changes of the coordinates coming from these displacements, consideredindependent of the cause of their generations. (1) This long paper (61 pages) studies motions in the 3-dimensional Euclidean space E systematically, starting from a system of axioms which imitates Euclid’s Elements, so tosay. Its highlight is to express the product of two spatial motions by calculation formulasobtained with the help of triangles on the sphere and expressed explicitly by means of trigono-metric functions. This calculation formulas give substantially the rule of the product in the quaternion algebra . (2) Another important contribution is
Rodrigues expression of a spatial rotation. Thisis understood, in recent years, more useful in practice than usual Euler angles. The reasonwhy is • (apart from Euler angles) it has no gaps (jumps) of parameters and accordingly differ-ential equations can be written with it easily, and multiple rotations beyond π (360 o ) canbe continuously written without problems, and • algorithms with less computational complexity can be given, etc. (3) Aside from these things, there appear here substantially the double covering of the ro-tation group SO (3) and its 2-dimensional spin representation (doubly-valued representation)as is explained below. A.1.1.2 More detailed explanation.
The original description of Rodrigues himself is mainly geometrical by using triangles ona sphere. But, here we explain the contents of a paper written about 180 years ago, in theworld of modern mathematics, so please permit us to use, for convenience, the quaternionitself and modern mathematical terminology. Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851) , i , j , k (see (1.4) below). The quatenion isgiven as H = R R i + Rj + Rk ĄCand the total set of pure quaternion numbers as H − = Ri + Rj + Rk ĄD The length of x = x + x i + x j + x k ( x j ∈ R ) is defined as k x k := p x + x + x + x , and the set of all quaternion numbers with length 1 is denotedby B , and we put B − := H − ∩ B . Any element of H − is expressed as φ w ( φ ∈ R , w ∈ B − ) .The following lemma gives elementary properties necessary in the following. Lemma A.1.1.
Put x := x − x i − x j − x k . (i) x x = k x k and, for x = 0 , we have x − = x / k x k . (ii) xy = y · x . (iii) k xy k = k x k · k y k ( x , y ∈ H ) . Proof.
By using (i) and (ii), (iii) is given as k xy k = ( xy ) xy = ( xy )( y · x ) = k x k · k y k . ✷ This lemma shows that H is a skew field, and the map x x reverses the order of theproduct. By the assertion (iii), B is closed with respect to the product, and since k u k = 1 gives k u − k = 1 , we know that B is a group. Lemma A.1.2. If we consider H as an algebra over R , then there exists an iso-morphism from its complexification H C = H ⊗ R C onto the full matrix algebra M (2 , C ) ofcomplex matrices of order 2. An isomorphism Φ is given by the correspondence: i → I = (cid:18) −
11 0 (cid:19) , j → J = (cid:18) ii (cid:19) , k → K = (cid:18) − i i (cid:19) . Proof.
By simple calculations, it is shown that the matrices
I, J, K satisfy the equationscorresponding to the so-called fundamental formula i = j = k = − , ijk = − . Togetherwith the unit matrix E , they form a basis over C of M (2 , C ) . ✷ A.1.1.3 Rodrigues expression of rotations.
We identify the 3-dimensional Euclidean sapce E with H − under the correspondence H − ∋ x = x i + x j + x k ←→ x = t ( x , x , x ) ∈ E ( x j ∈ R ) , where a point in E isdenoted with a column vector x . Since it can be proved that a rotation R arround the origin ∈ H − has necessarily an axis of rotation, it can be expressed by means of the axis ofrotation w ∈ B − and the angle φ ∈ R measured as a right-handed screw rotation. Here weaccept any angle, not restricted as | φ | ≤ π . Take the vector φ w ∈ H − as a parameter of R , This is a corrected version of Lemma A.1.2 in [Hir2]. Any rotation in odd-dimensional space necessarily has an axix of rotation. R as R = R ( φ w ) ( φ w ∈ H − ) . (1.1)This is called as Rodrigues expression of rotation.Now let us realize the rotation R ( φ w ) explicitly by means of φ and w . Lemma A.1.3.
The unit ball B in H is a group with respect to the product, and it givesa universal covering group of the rotation group SO (3) .Proof. For a u ∈ B , define a map Ψ( u ) : H − ∋ x x ′ = uxu − ∈ H − . Then, by Lemma A.1.1, k x ′ k = k u k · k x k · k u − k = k x k . Hence the corresponding map Ψ ′ ( u ) : E ∋ x x ′ = gx ∈ E conserves the length, that is, g = g ( u ) belongs to the length preserving group O (3) of E ∼ = H − . Moreover, g can be connected to the unit matrix E continuously, and so it belongsto the connected component SO (3) containing E , or it gives a rotation in E arround theorigin. The kernel of the homomorphism Ψ ′ : B ∋ u g ( u ) ∈ SO (3) is Ψ ′ − ( { E } ) = {± } ,and so B / {± } ∼ = SO (3) . Moreover, the 3-dimensional ball is simply connected, and so B issimply connected. Accordingly B is a universal covering group of SO (3) . ✷ Lemma A.1.4. (i)
For any x ∈ H − , there exists a u ∈ B such that x = u · θ i · u − with θ = k x k . (ii) The subset B − = B ∩ H − of B is a conjugacy class of B and B − = { x ∈ B ; x = − } .Proof. (i) As is well known, any element x ∈ E is sent to t ( θ, , with θ = k x k underthe action of SO (3) . By the correspondence H − ∼ = E and Ψ ′ : B ∋ u g ( u ) ∈ SO (3) , inthe proof of the previous lemma, pull back this to the world of H − and B .(ii) Apply (i) to an x ∈ B − . Then θ = k x k = 1 , and the element in H − correspondingto t (1 , , ∈ E is i . Hence x = uiu − ( ∃ u ∈ B ) . Also express x ∈ B as x = α + β i + γ j + δ k ( α, β, γ, δ ∈ R ) , then α + β + γ + δ = 1 . Apply here the condition x = − , then wesee that it is equivalent to the condition α = 0 . ✷ Lemma A.1.5.
For x ∈ H , the infinite series exp( x ) := X k< ∞ k ! x k s absolutely convergent, and for v ∈ H − , we have exp( v ) ∈ B or k exp( v ) k = 1 . Moreoverexpresse v as v = φ w ( w ∈ B − , φ ∈ R ) , then exp( φ w ) = cos φ + sin φ · w , exp (cid:0) ( φ + 2 kπ ) w ) = exp( φ w ) ( k ∈ Z ) .Proof. From P k< ∞ k k ! x k k = P k< ∞ k ! k x k k = exp (cid:0) k x k ) < ∞ , the absolute con-vergence follows. Also, from the expression v = φ w , by using w = − , we obtain exp( φ w ) = X p > ( − p (2 p )! φ p + X p > ( − p (2 p + 1)! φ p +1 w = cos φ + sin φ · w . ✷ Theorem A.1.6. (Rodrigues expression of a rotation)
Make the angle of rotation to ahalf as φ → φ and then take a composition map Ψ ′ ◦ exp : H − → SO (3) . This gives therotation R ( φ w ) ∈ SO (3) as H − ∋ φ w Ψ ′ (cid:0) exp( φ w ) (cid:1) = Ψ ′ (cid:0) cos( φ ) + sin( φ ) w (cid:1) ∈ SO (3) . Or the right hand side expresses the element in the rotation group SO (3) with rotation axis w and rotation angle φ .Proof. It is sufficient to prove the assertion in the case of w = i (Why ?). Denote by ρ the rotation Ψ ′ (cid:0) exp( φ i ) (cid:1) . Then, ρ ( i ) = i . Thus the rotation axis of ρ is i , and so ρ is expressed as a rotation in the ( j , k ) -plane perpendicular to i . By using the double angleformula for trigonometric function, we obtain (cid:0) ρ ( j ) , ρ ( k ) (cid:1) = ( j , k ) (cid:18) cos φ − sin φ sin φ cos φ (cid:19) . Hence ρ is a rotation in the ( j , k ) -plane with angle φ . ✷ Important remark.
Under the map Ψ ′ , the angle θ = φ in the parameter space H − appears twice as θ = φ for the rotation angle in E . This means that Ψ ′ : B ∋ u g = Ψ ′ ( u ) ∈ SO (3) is a map and the group B is a double covering group of SO (3) . Also, reverse the perspective andconsider the map (locally univalent but globally doubly-valued) as π : SO (3) ∋ g = g ( u ) u ∈ B ⊂ H . (1.2)Connect with this the isomorphism Φ : H ⊗ R C → M (2 , C ) , then there appears the 2-dimensional projective representation (the so-called spin representation ) Φ ◦ π of the rotationgroup SO (3) . 5 .1.1.4. Product of two rotations and calculation rule in quaternion. It is very natural from the above dicussions to imagine that, when the product R ( φ w ) R ( φ ′ w ′ ) ( w , w ′ ∈ B − , φ, φ ′ ∈ R ) of two rotations is expressed as R ( φ ′′ w ′′ ) , the rule of calculating ( φ ′′ , w ′′ ) from ( φ, w ) and ( φ ′ , w ′ ) comes out from the product rule in quaternion. (Note that, to modernize terminologiesand summarize the total story, we used quaternion from the beginning of the above discussons.We sincerely apporogise that the historical time goes back and forth as this.)In the above paper of Rodrigues, the word “quaternion” never appeared, but only geo-metric calculations concerning triangles on a sphere. However the calculation formula for thecomposition of two rotations are explicitly written out. This calculation rule is equivalent tothe following formula in the quaterinion H :(1.3) (cid:0) cos( φ ) + sin( φ ) w (cid:1)(cid:0) cos( φ ′ ) + sin( φ ′ ) w ′ (cid:1) = cos( φ ′′ ) + sin( φ ′′ ) w ′′ . Note.
Hoping to get a formula which is completely univalent, if we put some restriction tothe range of the variable φ in φ w , then some breaks of parameter φ will appear. Accordinglyit is better and natural to endure multivalency and take H − as the parameter space, and admitmultiple rotations arround a rotation axis w (Cf. [H5]). A.1.2. William Rowan Hamilton [Ham] W.R. Hamilton, On a new species of imaginary quantities connected with the theory ofquaternions,
Proc. Royal Irish Acad., (1843), 424–434. Starting from a certain point of his research life, Hamilton continued long time, as hisresearch objective, to extend complex field C = R R i, i = √− , to a vector space andmake it possible to divide a vector by another vector (he called this as a division by a vector).As the first step, he tried the case of adding two imaginary units to the real field R , and aftera considerable effort, he could succeeded finally, on the 16th of October 1843, to discoverthat, by adding three imaginary units, i , j , k , and introducing the following calculation rule,which Hamilton called fundamental formula , it is possible : i = j = k = − , ijk = − (fundamental formula).(1.4)After such a big discovery, firmly believing that “ the quaternion should be a great discoveryof the century ”, Hamilton was making efforts to show usefulness of the quaternion and toexpand the theory to the scientific world. However, he could not succeeded to express a William Rowan Hamilton (4 August 1805 – 2 September 1865) correspondence between the unit ball B ∼ = S of quaternion numbersand the total set SO (3) of 3-dimensional rotations.Actually, in the case of complex number field and 2-dimensional rotations, under thenatural correspondence C = R + R i ∋ x + yi ←→ t ( x, y ) ∈ E , identify C with the 2-dimensional Eucledean space E , then an element w = e iθ , ≤ θ < π, of B := { w ∈ C ; | w | = 1 } naturally corresponds 1 : 1 way to a 2-dimensional rotation of angle θ under thesimple multiplication as z = x + yi z ′ = e iθ z = x ′ + y ′ i , where (cid:18) x ′ y ′ (cid:19) = u ( θ ) (cid:18) xy (cid:19) , u ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) . Or, in the case of 2-dimensional rotations the natural isomorphism B ∋ e iθ → u ( θ ) ∈ SO (2) is realized. Remark A.1.2.1. Comentary from a point of view of the history of science.
We quote here a paper explaning a rather complicated situation from a point of view ofthe history of science arround Rodrigues, Hamilton and the quaternion. From this paper, Imyself got a lot of new important knowledge which I have never seen.[Alt] S.L. Altmann,
Hamilton, Rodrigues, and the quaternion scandal,
What went wrongwith one of the major mathematical discoveries of the nineteenth century , Mathematics Magazine, (1989), 291–308.There is a reality, which cannot be put aside under the simple word ‘ Irony of the History ,that the virtual discovery of quaternion of Rodrigues and his other important mathematicalworks were not taken into account and ignored totally.Historically, as a Jewish French, he suffered by the anti-Semitism of Cathoric CharchPower after Restoration of Imperial Rule something like ‘
Exclude Jews from public institu-tions ’, and so he could not find any teaching position and was obliged to be a banker as hisfamily business. But at his age over 40, he could publish the above excellent, long mathemat-ical paper [Rod], with author’s name Olinde Rodrigues, in Journal de Mathématiques Pureset Appliquées.Élie Cartan cited this paper in his book [Car2] on the theory of spinor ( spineur in French),on the subject of 2-dimensional spin representation of SO (3) or of so (3) , but he misunderstoodthat the paper is written by two coauthors called Olinde and Rodrigues respectively. In reality,the first name of Rodrigues is Benjamin and the middle name is Olinde, which is added inhis boyhood by his father after the official order of Cathoric Church Power that ‘ Everyoneshould have middle name ’, but this middle name
Olinde does not follow the cathoric styleand is very unusual (one of my French friends pointed out it to me). It may be a reason why7. Cartan mistook as two coauthors. Rodrigues himself used his middle name Olinde as thefirst name in his mathmatical papers.Before and after this paper, between 1838 and 1843, Rodrigues published 7 papers alto-gether in the same Journal. They contain short papers but every paper is good one withsharp cuts, I feel. He pulished them in between 43 and 48 ages, with the long silence over 20years after his university life.Why ? I don’t know, but these science-historical facts have been totally ignored untilrecently. However the mathematician Benjamin Olinde Rodrigues (1795–1851) now becomesknown and his honor is recovered (Cf. [AlOr]). Myself I studied him in many ways and madea report on B. O. Rodrigues in the 22nd symposium on the history of mathematics, 2011, heldat Tsuda University (Cf. [H5]).
A.2. History
A.2.1 Creation of theory of representations of groups [Fro1] F. Frobenius,
Über Gruppencharaktere,
Sitzungsberichte der Königlich Preußischen Akademieder Wissenschaften zu Berlin, , pp.985–1021.[Fro2] —,
Über die Primfactoren der Gruppendeterminante, ibid., , pp.1343–1382.[Fro3] —,
Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, ibid., ,pp.944–1015.
Dedekind was studying characters of non-abelian finite groups G , starting from 1880, andput some questions to Frobenius. From this occasin Frobenius has started his study on thetheory of characters of finite non-abelian groups and so on. The first results were publishedin papers [Fro1, 1896], [Fro2, 1896]. These are, above all, the beginning of the systematicstudies of the theory of linear representations (of finite groups). Just after, in the next paper[Fro3, 1897], the study of linear representations started and the relations to group charactersand group determinants were clarified.Frobenius defined in [Fro1] characters of non-abelian group by a system of equationsgiven by means of orders of several kinds of conjugacy classes, and gave a complete set of itssolutions. He studied in [Fro2] group determinant in a purely algebraic way. Next year, in[Fro3], he began to study actually linear representations of (finite) groups, and proved that thecharacter defined in [Fro1] is equal to the trace character of irreducible linear repreentations,and the group determinant in [Fro2] comes out from the regular representation on ℓ ( G ) andrules over, from an algebraic point of view, its decomposition into irreducible representations. Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) gave in [Bur1, 1898] – [Bur2, 1898] another way of approach,and Schur , a student of Frobenius, reconstruct the theory of linear representations andcharacters of a finite group in a modern style [Sch6, 1905]. [Mol2] T. Molien, Eine Bemerkung zur Theorie der homogenen Substitutionensgruppe,
Sitzungs-berichte der Dorpater Naturforscher-Gesellschaft, (1897), 259–274. Theodor Molien (Russian name, Fedor Eduardovich Molin) obtained his doctor’s degreeat Dorpat Imperial University, a frontier of German academic study of that time. The workof his thesis was published as [Mol1, 1892] and the main contribution is a proof of a part ofthe following Wedderburn’s Theorem in the case of the coefficient field K = C : A simple algebra over a commutative field K is isomorphic to the full matrixalgebra M ( n, D ) of a skew field D over K . Later he got a teaching position in that university, he wrote a paper [Mol2, 1897] in Bul-letin of Dorpat Natural Science Researcher’s Association, which contains an important resultproved by using the above result in [Mol1], independent of Frobenius. (Actually Frobeniusutilized the result of himself on associative algebra obtined erlier.) In §4 of [Fro3, 1897],Frobenius noticed this fact refering papers [Mol1] and [Mol2].A paper [Mol3, 1898] was published in Bulletin of Prussian Academy in Berlin under theintroduction by Frobenius. It seems that, afterwards Frobenius looked for some teachingposition for this young talented reseacher, but it did not succeed. Later, when Dorpat wasincluded in Soviet Union, Molien chose to remain there and was sent to Siberia and there hewrote many textbooks in mathematics (Cf. a historical study [H4]). William Burnside (2 July 1852 – 21 August 1927) Issai Schur (10 January 1875 – 10 January 1941) Theodor Molien (10 September 1861 – 25 December 1941) .2.2 Beginning of projective (or spin) representations ofgroups It was 1904, only 8 years later from the creation of the theory of linear repersentations ofgroups in 1896, that the theory of projective representations or spin representations of groupsbegun. [Sch1] J. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitu-tionen,
J. für die reine und angewante Mathematik, (1904), 20–50.[Title] On representations of finite groups through linear fractional transformations.
After his teacher Frobenius opened wide the gate to the thoery of linear representations of groups, Schur opend the gate to the theory of projective representations or spin represen-tatrions . The former is a homomorphism into GL ( n, K ) , K = C or R , and the latter is ahomomorphism into P GL ( n, K ) = GL ( n, K ) /K × . But Schur himself noticed at the top ofthis paper asDas Problem der Bestimmung aller endlichen Gruppen lineare Substitutionenbei gegebener Variabelnzahl n ( n > gehört zu den schwierigsten Problemen derAlgebra und hat bis jetzt nur für die binären und ternären Substitutionsgruppenseine vollständige Lösung gefunden. Für den allgemeinen Fall ist nur bekannt,daß die Anzahl der in Betracht kommenden Typen von Gruppen eine endliche ist;dagegen fehlt noch jede Übersicht über die charakteristischen Eigenschaften dieserGruppen.Die Umkehrung dieses Problems bildet in einem gewissen Sinne die Aufgabe:alle Gruppen von höchstens h ganzen oder gebrochenen linearen Substitutionenzu finden, die einer gegebenen endlichen Gruppe H der Ordnung h ein- ordermehrstufig isomorph sind, oder auch, wie man sagt, alle Darstellungen der Gruppe H durch lineare Substitutionen zu bestimmen.(Translation) The problem of determining all the finite groups of linear transformationsof given degree n ( n > belongs to the most difficult problems in algebra. And a completeanswer is obtained only for n = 2 , . In general cases, we know only that the number of typesof such finite groups is finite, and don’t have any prospect about characterizing properties ofsuch groups.If we think about it, this problem suggests the following research subject: Find all groups consisting of at most h number of linear transformations or of projectivetransformations, which is isomorphic or homomorphic to a given finite group H of order h .Or, in other words, determine all the linear or projective representations of H . determine all the finite groups contained in GL ( n, C ) .So the way of stating the results in this paper is that he put his weight to know the imagegroup π ( G ) ⊂ GL ( n, C ) of representation π of G , which is more stronger than the motivationto represent G or, so to say, the representation π itself as a map. Accordingly any symbolsdenoting representation itself (like π ) do not used.
1. Projective representation.
Here we give definition of projective representations,essentially same as that of Schur. A projective representation of a group G is a map π whichgives, for g ∈ G , a linear transformation π ( g ) over C in such a way that π ( g ) π ( h ) = r g,h π ( gh ) ( g, h ∈ G, r g,h ∈ C × ) , (2.5)and the function r g,h on G × G is called the factor set of π .On the other hand, a function r on G × G with values in C × := { z ∈ C ; z = 0 } satisfying r k,gh r g,h = r k,g r kg,h ( k, g, h ∈ G ) , (2.6)is called as a C × -valued 2-cocycle on G . The product of 2-cocycles is also a 2-cocycle,and the quotient of the set of all 2-cocycles modulo the equivalence relation r g,h ≈ r ′ g,h := r g,h · ( λ g λ h /λ gh ) , with λ g ∈ C × ( g ∈ G ) , is denoted by H ( G, C × ) and is called Schurmultiplier of G .In the paper [Sch1], the fundmental theory of projective representations of finite groupsare discussed. Nowadays, a central extention G ′ of G by a commutative group Z is definedby an exact sequence → Z → G ′ Φ → G → , (2.7)with a homomorphism Φ : G ′ → G , and Z ֒ → Z ( G ′ ) , the center of G ′ . Take in G ′ a section S of G as G ∋ g → s ( g ) ∈ S ⊂ G ′ , then s ( g ) s ( h ) = z g,h s ( gh ) ( g, h ∈ G, ∃ z g,h ∈ Z ) . (2.8)Now, for a linear representation Π of G ′ , put π ( g ) := Π (cid:0) s ( g ) (cid:1) ( g ∈ G ) . Then, for g, h ∈ G , π ( g ) π ( h ) = Π (cid:0) s ( g ) (cid:1) Π (cid:0) s ( h ) (cid:1) = Π (cid:0) s ( g ) s ( h ) (cid:1) = Π (cid:0) z g,h s ( gh ) (cid:1) = Π (cid:0) z g,h (cid:1) Π (cid:0) s ( gh ) (cid:1) = Π (cid:0) z g,h (cid:1) π ( gh ) , and so, when Π is irreducible, for z ∈ Z , we have Π( z ) = χ Z ( z ) I , by Schur’s Lemma , with I the identity operator. We call the character χ Z ∈ b Z of Z as spin type of Π (with implicitreference to the base group G ∼ = G ′ /Z ). Here Π( z g,h ) = r g,h I with r g,h = χ Z ( z g,h ) ∈ C × , andaccordingly π ( g ) π ( h ) = r g,h π ( gh ) ( g, h ∈ G ) , (2.9) 11nd so π is a projective representation of G ∼ = G ′ /Z with a factor set r g,h = χ Z ( z g,h ) .
2. Represetation group. A representation group G ′ of a finite group G is definedas a special central extension of G such that any projective representation π of G can beobtained as above from a linear representation Π of G ′ , and that the order | G ′ | is minimumamong central extensions having such property. Note that a projective representation of G is nothing but a multi-valued representation. In the paper [Sch1], among other things, thefollowing is shown: (1) For an arbitrary finite group G , there exist finite number of representation groups(modulo isomorphisms), (2) For any representation group G ′ of G , the commutative group Z in the central exten-sion (2.7) is isomorphic to Schur multiplier H ( G, C × ) of G . Note A.2.2.1.
In the case of a connected Lie group G , its universal covering groupscorrespond to the representation group in the case of finite groups. Taking a representationgroup of G , we denote it as R ( G ) . After the case of Lie groups, we call R ( G ) as a univrsalcovering group of G , even if it is not unique in general (Cf. A.2.3 , A.2.6 ). [Sch2] J. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochenelineare Substitutionen,
J. für die reine und angew. Mathematik, (1907), 85–137.[Title] Studies on the representations of finite groups by linear fractional transformations.
Here there were given methods of consturucting representation groups, evaluation of num-bers of non-isomorphic representation groups, and a method of calulating Schur multipliers.Furthermore, irreducible spin characters (characters of spin irreducible representations) wereexplicitly given for groups SL (2 , K ) , P SL (2 , K ) , GL (2 , K ) , P GL (2 , K ) , with K = GF [ p n ] .By this result, for these groups, the classification of irreducible spin representations werebasically completed. n thorder) [Sch4] J. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppendurch gebrochene lineare Substitutionen, ibid., (1911), 155–255.He constructed representation groups of the symmetric group S n of n -th order. Atually,for S n with n = 2 , , its representation group is S n itself, and for n ≥ , = 6 , the number of12on-isomorphic representation groups is 2ĄCand for n = 6 the number is 1. Also the type ofrepresentation groups of alternating group of n -th order is unique.Studying their linear representations, he gave spin representations of the base groups S n and A n . Their construction and the calculation of their characters were explicitly carried out.For the construction of spin representations, he first gave a fundamental spin representa-tion ∆ n called Hauptdarstellung of S n , utilizing almost the same × matrices as the so-calledPauli matrix triplets (1927) in A.2.4 below. Well managing ∆ n as a seed, he succeeded toconstruct all the irreducible spin representations, and then calculated their characters.In the calculation of these spin characters, there appear the so-called Q -functions of Schur.In total, his results are inclusive and even today has important influence. A.2.3 Cases of Lie groups and Lie algegras [Car1] É. Cartan,
Les groupes projectifs qui ne laissent invariante aucune multiplicité plane,
Bull.Soc. Math. France, (1913), 53–96.[Title] Groups of projective transformations which don’t leave any linear subspaces invariant. É. Cartan classifyed in this paper all the irreducible representation over C of any complexsimple Lie algebras. Here he utilized his classification of simple Lie algebras g over C , andproved that an irreducible representation of g is determined by its highest weight, and clarifiedhow the highest weight is determined.At that time, since the results on the structues of Lie group G corresponding to g and itsuniversal covering group are not sufficiently prepared, he did not so clearly recognize that, inthe classified irreducible representations, there are contained spin representations.For instance, the Lie algebra of 3-dimensional rotation group SO (3) is so (3) , isomorphicto su (2) whose corresponding Lie group is SU (2) . The latter is a universal covering group of SO (3) . The 2-dimensional natural representation SU (2) ∋ u u is, if seen from SO (3) , adoubly-valued representation, and this is the origin of the name of spin representation .If we see this from the level of Lie algebras, a complexification of so (3) is so (3 , C ) , whichis isomorphic to sl (2 , C ) . A universal covering group correspondig to the latter is SL (2 , C ) ,and its 2-dimensional natural representation SL (2 , C ) ∋ g g has the Lie algebra version sl (2 , C ) ∋ X X and an appropriate basis of sl (2 , C ) (triplet of matrices) is Pauli matirixtriplet, explained below.In the paper [Car1], regrettably enough, explicit matrix forms are not written, and so Élie Joseph Cartan, 9 April, 1869 – 6 May, 1951.
Expository paper: [CC] S.-S. Chern and C. Chevalley,
É. Cartan and his mathematicalwork,
Bull. Amer. Math. Soc., (1952), 217–250. A.2.4 Spin theory in quantum mechanics and mathematicalfoundation of quantum mechanics [Pau] W. Pauli,
Zur Quantenmechanik des magnetischen Elektrons,
Zeitschrift für Physik, (1927), 601–623.Let × Hermitian matrices be a = σ := (cid:18) (cid:19) , b = σ := (cid:18) − ii (cid:19) , c = σ := (cid:18) − (cid:19) , (2.10)then these are Pauli matrix triplet (or
Pauli matrices in short). Their commutation relationsare [ a, b ] = 2 ic, [ b, c ] = 2 ia, [ c, a ] = 2 ib ( i = √− . A.2.4.1 Covering map
Φ : SU (2) → SO (3) . Triplet { a, b, c } is a basis of sl (2 , C ) , isomorphic to the complexification of so (3) . Formore-mathematical expression, not containing i = √− in coefficients as above, take B j := iσ j ( j = 1 , , , then { B , B , B } is a basis over R of Lie algebra su (2) = Lie (cid:0) SU (2) (cid:1) , andthe commutation relations take the form [ B j , B k ] = 2 B l , (2.11)where ( j k l ) is a cyclic permutation of (1 2 3) .Now, let us give some more mathematical explanation. As noted a little in Subsection A.2.3 , a covering map
Φ : e G ∋ u g ∈ G from the universal covering group e G := SU (2) to the rotation group SO (3) is given as follows. For a column vector t ( x , x , x ) ∈ E , wemake correspond a × Hermitian matrix as X := X j x j σ j = (cid:18) x x − ix x + ix − x (cid:19) . (2.12) Wolfgang Ernst Pauli (25 April 1900 – 15 December 1958) u ∈ SU (2) , we give its action on X as X X ′ = uXu − = uXu ∗ and express itas X ′ = P j x ′ j σ j . Then, we obtain a linear transformation as ( x j ) j → ( x ′ j ) j .Its matrix expression gives the homomorphism g = Φ( u ) . By simple calculations, we get thefollowing formula (we leave the proof to readers): Φ (cid:0) exp( θB j ) (cid:1) = g j (2 θ ) (1 ≤ j ≤ , where exp( θB j ) , ≤ j ≤ , and g j ( ϕ ) , ≤ j ≤ , are in this order (cid:18) cos θ i sin θi sin θ cos θ (cid:19) , (cid:18) cos θ sin θ − sin θ cos θ (cid:19) , (cid:18) e iθ e − iθ (cid:19) , ϕ sin ϕ − sin ϕ cos ϕ , cos ϕ − sin ϕ ϕ ϕ , cos ϕ − sin ϕ ϕ cos ϕ
00 0 1 . In this case,
Ker(Φ) = {± E } and so Φ gives a double covering. Moreover, g j ( ϕ ) is thematrix of a rotation arround the central axis x j of angle ϕ , for j = 1 , , that advances theright-handed screw in the positive direction of the central axis, and for j = 3 , that advancesthe left-handed screw. Concerning one-parameter subgroup exp( θB j ) , if we take one cycle ≤ θ ≤ π of angle θ , then its image g j ( ϕ ) , ϕ = 2 θ, rotate two cycles (arround the axis x j ).The generators of one-parameter subgroup g j ( ϕ ) of G are A j := ddϕ g j ( ϕ ) (cid:12)(cid:12) ϕ =0 ∈ so (3) = so (3 , R ) ( j = 1 , , . (2.13)The set { A , A , A } gives a basis of so (3) , and has a system of commutation relations as [ A j , A k ] = A l , where ( j, k, l ) is a cyclic permutation of (1 , , . The differential d Φ gives anisomorphism B j → A j (1 ≤ j ≤ from mu (2) → so (3) . A.2.4.2 Two-dimensional spin representation of SO (3) . For g ∈ G = SO (3) , take a preimage u ∈ e G = SU (2) , Φ( u ) = g, of Φ and put π ( g ) := u, (2.14)then this is the 2-dimensional spin representation of the rotation group G . In reality, if g moves in a small nieghbourhood of the identity E , then g π ( g ) gives a 1 : 1 map (localisomorphism from G to e G ), but if ϕ moves continuously from θ to ϕ = θ + 2 π , then it arivesto π (cid:0) g j ( θ + 2 π ) (cid:1) = − π (cid:0) g j ( θ ) (cid:1) , that is, − appears as a multiplication factor. Thus it isnatural to accept this as a doubly-valued representation ( spin representation), or it can onlybe regarded as such.However when we step up from G to its universal covering group e G , it becomes univalentlinear representation (the identity representation u u ).15 maximal commutative subgroup (one of Cartan subgroups) of G = SO (3) is given by H = { g ( ϕ ) ; 0 ≤ ϕ < π } . (2.15)For a representation π on a representation space V ( π ) , take a common eigenvector v = 0 for the commutative family π ( h ) ( h ∈ H ) , and call it as a weight vector of π with weight χ ∈ b H , where π ( h ) v = χ ( h ) v ( h ∈ H ) .Ą@For h = g ( ϕ ) , there exists a k ∈ Z such that χ ( h ) = e ikϕ . Suppose π is irreducible.For π (one-valued) linear representation, k is integral ( k ∈ Z ) , and for (2-valued) spin rep-resentation, k is half an odd integer. The maximal one among these weights k is called the highest weight of π . We can prove by simple calculations that the highest weight determinesthe equivalence class of π completely.Note that the highest weight of the above two-dimensional π is equal to / . A.2.4.3 Original reason for the discovery of Pauli matrices.
The reason why Pauli arrived to the results in the above paper is the following:In the principle that “ Catch an electron as a mass point ”, the theory of applyingwave equation to a wave function gives double number as the number of stableorbits of an electron arround an atomic nucleus, and this phenomena is called duplexity phenomena . To avoid this contradiction, Pauli introduced the fourthquantum number, saying that electron has spin angular momentum of ± ~ .To say this mathmatically, “ Wave equation and wave function of electron receive transfor-mations not only by rotations in the space (as coordinate transformations) under the groupof rotations G = SO (3) but also transformations (through spin representation π ) under itsuniversal covering group e G = SU (2) . So it means that electron does not live in the 3-dimensional Eucledean space but in the 2-dimensionalspace C (or 4-dimensional space over R ) on which e G acts. This too much purely-mathmatical explanation could not be easily accepted by physisists,and for instance, one of my university class mates who studied physics said that, in the lecturecourse of physics in Departement of Physics, Kyoto University, Professor expalined in such away that, arround some axis (of which direction we don’t know), electron rotates .Detailed description and explanations about Pauli’s wave equation and wave functionsand how is the transformation of the group e G = SU (2) , are given in Chap. 4 of our book[HiYa]. [Dir] P.M.A. Dirac, The quantum theory of electron,
Proceedings of the Royal Society London A, (1928), 610-624; and Part II, ibid., (1928), 351–361.
16n this paper, Dirac enlarged a system of three matrices of Pauli to a system of four ma-trices and obtained spin representation of Lorentz group, and discovered the so-called Diracequation which is a relativistic (that is, Lorentz-invariant) wave equation. When electron isexpressed with this equation the above mentioned duplexy phenomena disappear automati-cally without any assumption.Moreover Klein-Gordon equation is factorized into two factors of ‘Dirac equations’, andthe latter is known to be much fundamental. Dirac equation gives a foundation of later bigdevelopment of quantum mechanics. To explain this in more detail, we first prepair somefundamentals about
Minkowski space M and Lorentz group L = SO (3 , . A.2.4.4 Minkowski space M . The Minkowski space M is a space to describe (space, time) consisting of (column)vectors x = t ( x , x , x , x ) with t ( x , x , x ) ∈ E , x = ct ∈ R , c = light velocity , t = time , and equipped with inner product given as h x , x i , := x + x + x − x = t x J , x , J , := diag(1 , , , − . The group O (3 , consists of all g ∈ GL (4 , R ) leaving the inner product invariant: h g x , g x i , = h x , x i , ( x ∈ M ) , or t gJ , g = J , . It has 4 connected components and the one containing the identity element e = E is called as proper Lorentz group and given as L := SO (3 ,
1) := (cid:8) g = ( g ij ) i,j ∈ O (3 ,
1) ; det( g ) = 1 , g ≥ (cid:9) . A.2.4.5 Dirac equation.
Dirac’s wave function ψ ( x ) is a function on M with values in V = C , written ascolumn vector as ψ ( x ) = (cid:0) ψ j ( x ) (cid:1) j . Dirac gave the so-called Dirac equation for elelctron, relativistic , as follows: put σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) , σ = (cid:18) (cid:19) , (2.16)and further put × matrices γ , γ , γ , γ as γ j = (cid:18) − iσ j iσ j (cid:19) (1 ≤ j ≤ , γ = (cid:18) − iσ iσ (cid:19) , (2.17) 17here denotes zero matrix of order 2. Then, in the case where the electromagnetic field iszero, Dirac equation is given, with the mass m of electron, as (cid:0) D + κ (cid:1) ψ = 0 , D := X j γ j ∂ x j , ∂ x j := ∂∂x j , κ := mc ~ . (2.18)The action of g ∈ L on the wave function is T g ( ψ )( x ) := g (cid:0) ψ ( g − x ) (cid:1) . Dirac equation is Lorentz-invariant or L -invariant, which is expressed as T − g (cid:0) D + κ (cid:1)(cid:0) T g ψ (cid:1) = (cid:0) D + κ (cid:1) ψ ( g ∈ L ) . Moreover, Klein-Gordon equation is factorized as ( (cid:3) − κ ) ψ = ( D − κ )( D + κ ) ψ = 0 , (cid:3) := X j ∂ x j − ∂ x . For more detailed explanation in this direction, please cf. [Ch. 5, HiYa] and [HHoH]. [Wey2] H. Weyl,
Gruppentheorie und Quantenmechanik , 1st edition, 1928 ; (2nd edition, 1931),Hirzel, Leipzig. English transl. of 2nd ed. by H. P. Robertson, Dutton, N.Y., 1932.[Neu] J. von Neumann,
Mathematische Grundlagen der Quantenmechanik , J. Springer, 1932.English transl. by R. T. Beyer:
Mathematical Foundations of Quantum Mechanics , PrincetonUniversity Press, 1955.
Following von Neumann’s idea (see [Neu], for instance) to describe the state of an ele-mentary particle, one utilizes a unit vector in a Hilbert space, or more exactly a complex line(Strahlenkörper) determined by the vector. Accordingly if a group acts on the space of states,then the action of this transformtion group is described by its
Strahldarstellung = rayrepresentation , which is essentially equal to a projective or spin representation. A.2.5 Rediscovery of a work of A.H. Clifford [Clif] A.H. Clifford,
Representations induced in an invariant subgroup,
Ann. Math., (1937),533–550. A.H. Clifford is an american mathematician, and his life time don’t overlap with that ofwell-known W.K. Clifford of Clifford algebra. He was an assistant of H. Weyl, at Princeton18nstitute for Advanced Study, when he wrote this paper. In it he studied in detail therestriction onto a normal subgroup N , of an irreducible representation of a finite group H . Hefound, in particular, in some cases, there appear inevitably projective (or spin ) representationsas tensor product factors.The following theorem is a result of summarizing some essential parts in §§3–4 of thispaper in my own way. Theorem A.2.5.1. (cf. [Clif, §§3–4])
Let N be a normal subgroup of a finite group H .For an irreducible linear representation τ of H , suppose that all the irreducible componentsof the restriction τ | N are mutually equivalent, that is, τ | N ∼ = [ ℓ ] · ρ ,where ρ is an irreduciblerepresentation of N , and ℓ = dim τ / dim ρ is its multiplicity. Moreover assume that the basefield is algebraically closed. (i) Irreducible representation τ of H is equivalent to a tensor product of two irreducibleprojective matrix representations C and Γ : (2.19) τ ( h ) ∼ = Γ( h ) ⊗ C ( h ) ( h ∈ H ) , where the factor sets of C and Γ are mutually inverse of the other, and dim Γ = ℓ, dim C = dim ρ,ρ ( h − uh ) = C ( h ) − ρ ( u ) C ( h ) ( h ∈ H, u ∈ N ) . (ii) It can be normalized as follows: for h ∈ H, u ∈ N , C ( hu ) = C ( h ) ρ ( u ) , C ( u ) = ρ ( u ) ; Γ( hu ) = Γ( h ) , Γ( u ) = E ℓ . In this situation, h Γ( h ) is substantially a projective representation of the quotient group H/N , and the factor sets of C and Γ are mutually inverse of the other, and they are sub-stantially factor sets of H/N . The result of Clifford has an important meaning for the stand point of spin representationsin the theory of group representations. It says that evenwhen we are working only on linear representations of groups, there may appear inevitably spin representations of some subgroupsin the process of constructing all irreducible linear representations.One simple such example is the case of semidirect product groups H = U ⋊ S in [Hir1],where U is compact, normal in H , and S is finite. In [Hir1] we gave a method of constructing aset of complete representatives of the dual b H by using spin representations of certan subgroupsof S . Our proof there is independent of the result of Clifford but we can give another proofof it by utilizing Theorem A.2.5.1 above.Note that, in the work of G.W. Mackey [Mac1]–[Mac2], S is assumed to be abelian and sothere do not appear any spin representations. The above classical results of Clifford is alsolinked to our recent works [HHH4], [HHo] and [Hir2].19 ote A.2.5.1. Now let me give a short break, intermediate. Then I would like tostate some personal thought about spin representations . By an invitation of late Prof. MitsuoSugiura, I joined to series of anual meetings held at Tsuda University as “ Symposium onHistory of Mathematics ”. In the first year, I reported on my work on representations andcharacters of semisimple Lie groups as a part of mathematics at present. But starting fromthe next year, I reported on a series of meomorial works of Frobenius which originated thetheory of linear representations and chracters of (finite) groups, according to the years ofpublication successively as in [H1], consecutively in four years.After that, I begun to follow works of Schur, a student of Frobenius. Papers of his firststage are some kind of extention or simplifiction of the results of his teacher. Then, at thesecond stage, I met the difficult work, a triplet of papers ( [Sch1, 1904], [Sch2, 1907], [Sch4,19011]) , on spin (or projective or multi-valued ) representations of groups, in the terminologyof Schur, Darstellung durch gebrochene lineare Substitutionen , and I made 2 reports on thesepapers in [H2].Then my interest spreaded naturally over the surrounding area containing the case of Liegroups, and I felt curious about something. In the 3rd paper [Sch4] of Schur, there appearedfully, many interesting results on spin (or projective) representations of symmetric groups S n and alternating groups A n . However, after that, there is a significant long, blank periodbefore some succeeding results were published on finite groups , whereas, for semisimple Liegroups including motion groups and Lorentz groups, the theory of spin representations wasstudied continuously and steadily.For instance, after almost half a century of blank, there finally appear Morris’ paper [Mor1,1962] on spin representations of symmetric groups, and Ihara-Yokonuma’s paper [IhYo, 1965]on Schur multipliers M ( G ) for finite (and also infinite) reflexion groups G . For these historicalphenomena, there might be some reason, but I felt that the theory of spin representations inthis area are treated in some sense as a stepchild of the main theory of representations.However, in the above work of A.H. Clifford, it is proved that, in the theory of linearrepresentations of G , there appears naturally and inevitably spin (or multi-valued or projective )representations of some subgroups of G . This shows that spin representations are nothingbut a legitimate child of the theory of group representations. A.2.6 Developments in Mathematics, and QuantumMechanics [Wey1] H. Weyl,
Classical Groups, Their Invariants and Representations , Princeton UniversityPress, 1939.
Classical groups (or Lie groups of classical type) over C are given, up to isomorphisms,20s follows: type A n SL ( n +1 , C ) ( n ≥ , type B n SO (2 n +1 , C ) ( n ≥ , type C n Sp (2 n, C ) ( n ≥ , type D n SO (2 n, C ) ( n ≥ , and their compact real forms are respectively SU ( n +1) , SO (2 n +1) , Sp (2 n ) = SU (2 n ) ∩ Sp (2 n, C ) , SO (2 n ) . By means of the so-called
Weyl’s unitarian trick , irreducible unitary representations of thelatter corresponds 1-1 way to irreducible (finite-dimensional) holomorphic representations ofthe former. These representations are all constructed at least in principle and their charactersare explicitly calculated.The compact groups SU ( n + 1) and Sp (2 n ) are simply connected together with theirmother groups SL ( n +1 , C ) and Sp (2 n, C ) . Whereas SO ( N ) , N ≥ , has (double covering)universal covering group Spin( N ) and so they have irreducible linear representations and alsoirreducible spin (or projective , or doubly-valued ) representations. But they can be treated ina same way, and have a unified common formula for characters and dimensions for each type. SL (2 , R ) , double coveringof 3-dimensional Lorentz group) [Bar] V. Bargmann, Irreducible unitary representations of the Lorentz group,
Ann. Math., (1947),568–640. A double covering group of 3-dimensional Lorentz group L = SO (2 , , with the 2-dimensional space-part, is realized by SL (2 , R ) . Bargmann constructed all irreducible unitary(and also non-unitary) representations SL (2 , R ) and gave their characters. Seeing from thebase group L , a half of these representations is linear and another half is of spin (doubly-valued). A universal covering group of L is ∞ -times covering and the situation differs verymuch from the case of SL (2 , R ) . SL (2 , C ) , a universal covering group of4-dimensional Lorentz group) [GeNa] I.M. Gelfand and M.I. Naimark, Unitary representations of Lorentz group (in Russian),Izvestia Akad. Nauk SSSR, (1947), 411–504 [ English Translation in Collected Works of Gelfand,Vol. 2, pp.41–123 ]. The universal covering group SL (2 , C ) of 4-dimensional Lorentz group L is doubly cov-ered. In this paper, all irreducible representations are constructed and their characters arecalculated explicitly. Seeing from the base group L = SO (3 , , there are linear representa-tions and spin representations, and they can be treated in a similar way.21
947 (Infinitesimal construction of irreducible representations of 4-dimensionalLorentz group) [HC] Harish-Chandra (Doctor’s Thesis),
Infinite irreducible representations of the Lorentz group,
Proceedings of the Royal Society London A, (1947), 327–401.
In colonial ages of India, an excellent local academic person could be sent to colonialmother country, Great Britain, for much advanced studies. Thus Harish-Chandra studied inLondon for his Doctrate under the supervision of Dirac.Anyhow, the subject of this work was suggested to him by Dirac as is noted in its Introduc-tion. He classified and constructed irreducible representations of 4-dimensional Lorentz group L = SO (3 , on the level of Lie algebras. This is called as infinitesimal method. Since thisis the story on the level of a Lie algebra, there didn’t appear the theory of characters, whichare special functions on the corresponding Lie group. A.2.7 Renaissance of the theory of spin representations offinite groups
For a connected semisimple Lie group, when we treat finite dimensional irreducible repre-sentations, their highest weights have some differences between the cases of linear represen-tations and spin (multi-valued) representations, but in principle they can be treated equallyin a unified way.For example, for the rotation group SO (3) , the highest weights of the former are non-negative integers, whereas those for doubly valued spin representations are positive half inte-gers. However the character formula and the dimension formula take the same form and haveno difference. And passing through the ages of É. Cartan, J. Schur (= I. Schur), and H. Weyl,of finite-dimensional representations, we came to such ages as treating infinite-dimensionalrepresentations of Lorentz groups and their covering groups SL (2 , R ) and SL (2 , C ) in anatural unified way, not so much depend on linear or spin , and so on.On the contrary, for finite groups, linear representations and spin (or multi-valued ) onesare, in many cases, have quite different features. Maybe depending on this fact partly, andalso depending partly on that Schur worked out in his trilogy [Sch1], [Sch2], and [Sch4], toocompletely, comprehensively and thoroughly, so that his successors could not appear for longtime. [Mor1] A.O. Morris, The spin representation of the symmetric group,
Proc. London Math. Soc., (3) 12 (1962), 55–76.
After half a century of Schur’s trilogy, there opened the study of spin representations. Theabove is the first of such papers. Its content begun with recapturing Schur’s theory. Later,22tudents of Morris and other peoples gradually follow spin theory of finite groups. H ( G, C × ) ) [IhYo] S. Ihara and T. Yokonuma, On the second cohomology groups (Schur multipliers) of finitereflexion groups,
J. Fac. Sci. Univ. Tokyo, Ser.1, IX (1965), 155–171. From a more algebraic point of view, the determination of Schur multiplier H ( G, C × ) isthe theme studied hardly and intensively in some period of time. According to Gorenstein’sbook [Chap.2, Gor], Schur multiplier is treated as an important datum for known finite groupsto prepare a complete classification of finite simple groups, and this gives partly a motivationof calculating H ( G, C × ) explicitly.However, from the explicit determination of H ( G, C × ) to the explicit determination ofall irreducible spin (multi-valued) representations, they didn’t proceed immediately to thissecond step. There was certain time gap. Why ? I don’t know, but is there some psychologicalbarrier ? Here I quote early three papers. [Mor2] A.O. Morris,
Projective representations of abelian groups,
London Math. Soc., (2)7 (1973), 235–238. [DaMo] J.W. Davies and A.O. Morris,
The Schur multiplier of the generalized symmetricgroup,
J. London Math. Soc., (2) 8 (1974), 615–620. [Rea] E.W. Read,
On the Schur multipliers of the finite imprimitive unitary reflexion groups G ( m, p, n ) , J. London Math. Soc., (2), 13 (1976), 150–154.
A.2.8 Weil representations of symplectic groups [Wei2] A. Weil,
Sur certains groupes d’opérateurs unitaires,
Acta Math., (1964), 143–211.
As for the motivation of this work, maybe Weil had as his purpose cetain images of workingon algebraic groups over adele groups. Here, more generally, he uses irreducible representa-tions of Heisenberg groups over locally compact abelian groups and groups consisting of theirintertwining operators. These groups give spin representations of symplectic groups, which iscalled
Weil representations . Once I wrote a letter to Prof. Yokonuma thanking his two papers on Schur multipliers of finite and infinitereflexion groups. At that occasion I put a similar question like “ Why don’t you proceed, after determinationof Schur multipliers, to construction process of multi-valued irreducible representations, taking the datum H ( G, C × ) for help ? ” His answer is somewhat vague and means something like “ If I have such an idea, thenit would be much better for me.”
23n the special case where the locally compact abelian group is simply real number field R ,the symplectic group appeared is just Sp (2 n, R ) and the group of intertwining operators is itsdouble covering group, called Metaplectic group and denoted as Mp (2 n, R ) . Actually, Weilrepresentation in this case is a doubly-valued representation of the base group Sp (2 n, R ) ,which turns out to be a linear representation of Mp (2 n, R ) if going up to the upper level. [Sait] M. Saito, Représentations unitaires des groupes symplectiques,
J. Math. Soc. Japan, (1972), 232–251. A maximal compact subgroup of Sp (2 n, R ) is isomorphic to U ( n ) . Their universal cov-ering groups f Sp ( n, R ) and e U ( n ) are both infinite times covering. For any positive integer k , there exists (exactly) k -times covering group. Weil representation is doubly-valued for Sp (2 n, R ) , and linear for Mp (2 n, R ) . M. Saito constructed several series of irreducible uni-tary representations, using Weil representation : f Sp (2 n, R ) the universal covering group ↓ Mp (2 n, R ) double covering group ↓ Sp (2 n, R ) base group [Yos1] H. Yoshida, Weil’s representations of the symplectic groups over finite fields,
J.Math. Soc. Japan, (1979), 399–426. For the symplectic group Sp (2 n, K ) over a finite field K , its Weil representation is anirreducible linear representations of M p (2 n, K ) . [Yos2] H. Yoshida, Remarks on metaplectic representations of SL (2) , J. Math. Soc. Japan, (1992), 351–373. For a local field with characteristic = 2 , consider n -times covering group e G of G = SL (2 , K ) . For n = 2 and K = C , Weil representation of G can be lifted up to an irreduciblelinear representation of e G . In this paper, for any n ≥ , Yoshida constructed spin irreduciblerepresentations of G which becomes linear if going up to exactly n -times covering e G . References [Alt] S.L. Altmann,
Hamilton, Rodrigues, and the quaternion scandal,
What went wrong withone of the major mathematical discoveries of the nineteenth century , Mathematics Magazine, (1989), 291–308. AlOr] S. Altmann and E. Ortiz edit.,
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